A232600 a(n) = Sum_{k=0..n} k^p*q^k, where p=1, q=-2.

0, -2, 6, -18, 46, -114, 270, -626, 1422, -3186, 7054, -15474, 33678, -72818, 156558, -334962, 713614, -1514610, 3203982, -6757490, 14214030, -29826162, 62448526, -130489458, 272163726, -566697074, 1178133390, -2445745266, 5070447502, -10498808946, 21713445774

Offset: 0

Author

Stanislav Sykora, Nov 27 2013

Keywords

• sign
• easy

Examples

			a(3) = 0^1*2^0 - 1^1*2^1 + 2^1*2^2 - 3^1*2^3 = -18.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..1000
• S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
• Index entries for linear recurrences with constant coefficients, signature (-3,0,4).

Crossrefs

Cf. A045883, A140960 (absolute values), A059841 (p=0, q=-1), A130472 (p=1 ,q=-1), A089594 (p=2, q=-1), A232599 (p=3, q=-1), A126646 (p=0, q=2), A036799 (p=1, q=2), A036800 (p=q=2), A036827 (p=3, q=2), A077925 (p=0, q=-2), A232601 (p=2, q=-2), A232602 (p=3, q=-2), A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2).

Cf. A045883.

Programs

Magma

[2*((-2)^n*(3*n+1) -1)/9: n in [0..30]]; // G. C. Greubel, Mar 31 2021

Maple

A232600:= n-> 2*((-2)^n*(3*n+1) -1)/9; seq(A232600(n), n=0..30); # G. C. Greubel, Mar 31 2021

Mathematica

Table[2((3n+1)(-2)^n -1)/9, {n, 0, 30}] (* Bruno Berselli, Nov 28 2013 *)

PARI

a(n)=-((3*n+1)*(-2)^(n+1)+2)/9;

Sage

[2*((-2)^n*(3*n+1) -1)/9 for n in (0..30)] # G. C. Greubel, Mar 31 2021

Formula

a(n) = 2*( (3*n+1)*(-2)^n - 1 )/9.

abs(a(n)) = 2*A045883(n) = A140960(n).

From Bruno Berselli, Nov 28 2013: (Start)

G.f.: -2*x / ((1 - x)*(1 + 2*x)^2). [corrected by Georg Fischer, May 11 2019]

a(n) = -3*a(n-1) +4*a(n-3). (End)

From G. C. Greubel, Mar 31 2021: (Start)

E.g.f.: (2/9)*(-exp(x) + (1-6*x)*exp(-2*x)).

a(n) = 2*(-1)^n*A045883(n). (End)